| L(s) = 1 | + (0.849 − 2.61i)2-s + (0.0476 + 0.757i)3-s + (−4.50 − 3.27i)4-s + (1.44 − 0.182i)5-s + (2.02 + 0.519i)6-s + (−1.39 − 2.96i)7-s + (−7.93 + 5.76i)8-s + (2.40 − 0.303i)9-s + (0.749 − 3.92i)10-s + (−0.289 + 4.59i)11-s + (2.26 − 3.56i)12-s + (−0.448 − 0.542i)13-s + (−8.93 + 1.12i)14-s + (0.206 + 1.08i)15-s + (4.89 + 15.0i)16-s + (2.16 − 4.60i)17-s + ⋯ |
| L(s) = 1 | + (0.601 − 1.84i)2-s + (0.0275 + 0.437i)3-s + (−2.25 − 1.63i)4-s + (0.645 − 0.0815i)5-s + (0.825 + 0.212i)6-s + (−0.526 − 1.11i)7-s + (−2.80 + 2.03i)8-s + (0.801 − 0.101i)9-s + (0.237 − 1.24i)10-s + (−0.0871 + 1.38i)11-s + (0.653 − 1.03i)12-s + (−0.124 − 0.150i)13-s + (−2.38 + 0.301i)14-s + (0.0534 + 0.280i)15-s + (1.22 + 3.76i)16-s + (0.525 − 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 + 0.597i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.801 + 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.436419 - 1.31632i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.436419 - 1.31632i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 151 | \( 1 + (-11.5 - 4.19i)T \) |
| good | 2 | \( 1 + (-0.849 + 2.61i)T + (-1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (-0.0476 - 0.757i)T + (-2.97 + 0.375i)T^{2} \) |
| 5 | \( 1 + (-1.44 + 0.182i)T + (4.84 - 1.24i)T^{2} \) |
| 7 | \( 1 + (1.39 + 2.96i)T + (-4.46 + 5.39i)T^{2} \) |
| 11 | \( 1 + (0.289 - 4.59i)T + (-10.9 - 1.37i)T^{2} \) |
| 13 | \( 1 + (0.448 + 0.542i)T + (-2.43 + 12.7i)T^{2} \) |
| 17 | \( 1 + (-2.16 + 4.60i)T + (-10.8 - 13.0i)T^{2} \) |
| 19 | \( 1 + (-4.59 - 3.33i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-2.09 - 1.51i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-3.50 - 1.38i)T + (21.1 + 19.8i)T^{2} \) |
| 31 | \( 1 + (0.950 + 0.892i)T + (1.94 + 30.9i)T^{2} \) |
| 37 | \( 1 + (3.74 + 4.53i)T + (-6.93 + 36.3i)T^{2} \) |
| 41 | \( 1 + (3.44 + 0.885i)T + (35.9 + 19.7i)T^{2} \) |
| 43 | \( 1 + (2.95 - 6.28i)T + (-27.4 - 33.1i)T^{2} \) |
| 47 | \( 1 + (7.93 + 2.03i)T + (41.1 + 22.6i)T^{2} \) |
| 53 | \( 1 + (1.72 + 2.72i)T + (-22.5 + 47.9i)T^{2} \) |
| 59 | \( 1 + (-2.85 - 8.79i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (0.730 - 11.6i)T + (-60.5 - 7.64i)T^{2} \) |
| 67 | \( 1 + (4.45 + 0.562i)T + (64.8 + 16.6i)T^{2} \) |
| 71 | \( 1 + (6.09 + 12.9i)T + (-45.2 + 54.7i)T^{2} \) |
| 73 | \( 1 + (-4.13 - 8.78i)T + (-46.5 + 56.2i)T^{2} \) |
| 79 | \( 1 + (1.96 - 1.84i)T + (4.96 - 78.8i)T^{2} \) |
| 83 | \( 1 + (7.88 - 4.33i)T + (44.4 - 70.0i)T^{2} \) |
| 89 | \( 1 + (-6.13 + 3.37i)T + (47.6 - 75.1i)T^{2} \) |
| 97 | \( 1 + (9.42 + 1.19i)T + (93.9 + 24.1i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.59457854983922430126903442917, −11.69501329114303332354964462122, −10.31431402598348779401055073632, −9.947703204158636194969575746360, −9.420904302476087439690598926978, −7.22997455399488973502395119992, −5.31889641101115198890810802422, −4.35568502218082690888122499059, −3.24155691502905623958895639453, −1.48372083747406084777725157887,
3.30656824447521359658562607098, 5.09581517585492588509932583954, 6.03254152370744062325568359075, 6.68996323245046991013333050086, 8.000536726875607495136740129762, 8.839050705707618651321166609546, 9.877401818407238801640813542688, 11.98352420612573266294801192883, 12.93249156787556538848352224007, 13.54836028887661872745646114021
